My pumpkin vines are taking off and growing out of the raised bed. I was expecting that. I'm OK with it growing on the lawn -- I was going to mow one more time then let the pumpkin grow, but it occurred to me that it might be better if I got some straw and layered flakes of straw over the grass (maybe with paper sheet mulch underneath) for the pumpkin vine to grow over. I'm planning to convert this area to another raised bed anyway so it's perfectly fine to let the grass die. There IS a patch of clover that I might want to save but it's not a big deal.

Is it better for the pumpkin vines to grow on living grass (but the grass would grow tall between the leaves and possibly shade them?) or on straw mulch?

I think it is a matter of choice. As you know pumpkin vines and leaves are pretty large. I have had volunteers come up in my yard and I let them roam - had pretty pumpkins as a result. Let us know how you fare.

Two years ago, I threw some dead pumpkins out in the leaf pile that I had gathered under a set of three butternut trees, with the intention of letting it rot off during the fall/winter months. It did, for the most part, and the remainder was burned along with the leaves in that spring. I have since removed the trees, and have a rather large (approx. 12'x12'x10') brush pile laying here, that I've been using gradually to fuel campfires. A short distance away, I have some "campfire logs" - mostly 2"-3" branches cut to use for backyard fires.

...Both piles are COVERED in pumpkin vines. I mean, blanketed. No flowers yet, but I wouldn't be shocked to get some accidental baby pumpkins to harvest by October. Funny how these things work sometimes. =)